Coherent Control of Total Transmission of Light through Disordered Media

We demonstrate order of magnitude coherent control of total transmission of light through random media by shaping the wave front of the input light. To understand how the finite illumination area on a wide slab affects the maximum values of total transmission, we develop a model based on random matrix theory that reveals the role of long-range correlations. Its predictions are confirmed by numerical simulations and provide physical insight into the experimental results.

Figure 1

(a) Schematic of the experimental setup for the control of total transmission. As detailed in the Supplemental Material [22], the two polarizations of a Nd:YAG laser, $\lambda =532\mathrm{nm}$, are modulated by two different areas of a phase-only SLM. The scattering sample is placed at the focal plane of the objective. Three photodetectors, ${\mathrm{PD}}_1$, ${\mathrm{PD}}_2$, and ${\mathrm{PD}}_3$, measure, respectively, the intensities of transmitted, incident, and reflected light. (b) Measured $T/⟨T⟩$ (left panel) and $R/⟨R⟩$ (right panel) versus the optimization step for enhancement (increasing blue curve) and reduction (decreasing red curve) of the total transmission. The sample is $20\mu \mathrm{m}$ thick, and the average transmission $⟨T⟩\sim 5\%$. The dotted line represents the reflection estimated from the transmission using $R/⟨R⟩=(1-T)/(1-⟨T⟩)$.

Figure 2

Enhancement of transmission as a function of the sample thickness $L$ for an illumination diameter of $D=8.3\mu \mathrm{m}$ (a) and as a function of the input illumination diameter $D$ for a fixed sample thickness $L=23\mu \mathrm{m}$ (b). Experimental data are shown by blue dots, the estimation by Eq. (1) in red dotted lines. (c) $T^{\max }/⟨T⟩$ as a function of the fraction of controlled input channels $N_{\text{in}}/N_{\text{tot}}$. Experimental data are blue dots with the error bars from ensemble measurements; the dotted red curve represents the prediction of Eq. (1) for uncorrelated systems. (d) Representative amplitude patterns of the SLM macropixels for three illumination diameters $D$. The dark macropixels are switched off.

Figure 3

(a),(b) Geometries for which the fraction of controlled channels $m_1$ is given by a simple analysis of the input field (see text). (c) Effective fraction of controlled channel (2) in the geometry relevant for our optical experiment. Numerical results (dots) are obtained from the simulation of the wave equation in a two-dimensional disordered slab, for different values of the illumination diameter $D$ and the slab thickness $L$. The dielectric function is $\epsilon (\mathbf{r})=n_0^2+\delta \epsilon (\mathbf{r})$, with $n_0=1.3$ and $\delta \epsilon (\mathbf{r})$ uniformly distributed between [$-1.02$, 1.02] in the slab and $\delta \epsilon (\mathbf{r})=0$ outside the slab. The four sets of points correspond to $kL=120$, 187, 280, 450. The solid lines represent the theoretical prediction (3), where $I(\mathbf{q})=D\text{sinc}(qD/2)$.

Figure 4

Maximal transmission in a slab of thickness $L$, illuminated with $D=16\lambda$. (a) 2D simulations (dots), calculated with the same parameters as in Fig. 3, are compared with the theoretical prediction (solid and dashed lines) based on Eqs. (4) and (3). The effect of the finite number of modes is taken into account, as detailed in the Supplemental Material [22], and the effect of the sequential algorithm (blue squares) is included in the theory through the substitution $m_1\to \alpha m_1$, with $\alpha \simeq 0.26$. (b) The 3 D experiment (dots) is well described by Eq. (4), where $m_1\to \alpha m_1$, with $m_1$ given by Eq. (3) solved in 3D and $\alpha$ identical to the 2D case.